Classical WENO Schemes Research Articles

High-order finite volume multi-resolution WENO schemes with adaptive linear weights on triangular meshes

This paper presents high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights to solve hyperbolic conservation laws on triangular meshes. They are abbreviated as the ALW-MR-WENO schemes. The novel third-order, fourth-order, and fifth-order ALW-MR-WENO schemes are designed by applying two unequal-sized hierarchical central stencils in comparison to the classical WENO schemes which utilize many equal-sized upwind biased/central stencils. With the application of one simple condition, only two linear weights are automatically adjusted to be positive values on condition that their summation is one. The novel finite volume ALW-MR-WENO schemes could maintain the designed order of accuracy in smooth areas and reduce to the first-order accuracy so as to keep essentially non-oscillatory properties around strong discontinuities. So it is the first time that any high-order WENO schemes with the application of only two unequal-sized stencils are obtained on triangular meshes. And the major benefits are their efficiency, compactness, and simplicity in large scale engineering applications on unstructured meshes. Finally, several tests are used to indicate the effectiveness of these new finite volume WENO schemes.

Compact high-order gas-kinetic scheme for direct numerical simulation of compressible turbulent flows

In this paper, the three-dimensional fully compact high-order gas-kinetic scheme (HGKS) is proposed for the direct numerical simulation of compressible turbulent flows. Because of the high-order gas evolution model, the numerical fluxes as well as the point-wise conservative variables can be evaluated from the time-accurate gas distribution function at the cell interface. As a result, both the cell-averaged variables and their cell-averaged gradients can be updated inside each cell. Based on the cell averaged values and their gradients, the compact Hermite weighted essentially non-oscillatory (HWENO) scheme is developed, in which the dimension-by-dimension reconstruction is used for three-dimensional turbulences. In both normal and tangential directions, the fifth-order HWENO reconstruction is adopted. Compared with the classical WENO scheme, the stencil for the HWENO scheme only contains 33 cells for each cell. To achieve the temporal accuracy, the two-stage fourth-order temporal discretization is used. For the evaluation of point-wise variables, the simplified third-order gas-kinetic solver is used. Several classical benchmark problems are simulated, which validate the accuracy, resolution, and robustness of compact HGKS. As a comparison, the numerical results of HGKS using non-compact WENO reconstruction are also provided. Due to the compact stencil, the compact HGKS has a favorable performance for turbulence simulation in resolving the multi-scale structures.

Affine-invariant WENO weights and operator

The novel and simple nonlinear affine-invariant weights (Ai-weights) are devised for the Ai-WENO operator to handle the case when the function being reconstructed undergoes an affine transformation (Ai-operator) with a constant scaling and translation (Ai-coefficients) within a global (WENO) stencil. The Ai-weights essentially decouple the inter-dependencies of the Ai-coefficients and sensitivity parameter effectively. For any given sensitivity parameter, the Ai-WENO operator guarantees that the WENO-reconstructed affine-transformed-function remains unchanged as the affine-transformed WENO-reconstructed-function. In other words, the Ai-operator commutes with the nonlinear WENO operator (Ai-property) as proven theoretically and validated numerically. With a small scaling and a non-zero translation, the Ai-WENO scheme with a typical sensitivity parameter satisfies the ENO-property even when the corresponding classical and scale-invariant WENO scheme does not. In solving the shallow water wave equations and the Euler equations under gravitational fields, the characteristic-wise well-balanced Ai-WENO scheme satisfies the well-balanced property intrinsically without imposing the WENO linearization technique. Any Ai-weights-based WENO operator enhances the robustness and reliability of the WENO scheme for solving hyperbolic conservation laws.

A new finite difference mapped unequal-sized WENO scheme for Hamilton-Jacobi equations

In this paper, a new fifth-order finite difference mapped unequal-sized weighted essentially non-oscillatory (MUS-WENO) scheme is proposed for solving Hamilton-Jacobi equations in multi-dimensions. This new MUS-WENO scheme uses the same widest spatial stencils as that of the classical same order finite difference WENO scheme [19], and could obtain smaller truncation errors and optimal fifth-order accuracy in smooth regions. The MUS-WENO scheme uses a convex combination of a quartic polynomial with two quadratic polynomials. The linear weights can be artificially set as any positive constants on condition that their summation is one. Together with a new mapping function, associated new mapped nonlinear weights are proposed to reduce the difference with the linear weights. So it is a first time that the new finite difference MUS-WENO scheme with a very small ε could reach optimal accuracy order convergence even near extreme points in smooth regions and avoid spurious oscillations near strong discontinuities when solving Hamilton-Jacobi equations. Generally speaking, the main advantages of such new MUS-WENO scheme comparing with the classical WENO scheme [19] are its good convergence, robustness, efficiency and easy extension to multi-dimensions. Numerical experiments are proposed to show the good performance of this new MUS-WENO scheme.

A New Sixth‐Order Finite Difference Compact Reconstruction Unequal‐Sized WENO Scheme for Nonlinear Degenerate Parabolic Equations

In this paper, a new sixth‐order finite difference compact reconstruction unequal‐sized weighted essentially nonoscillatory (CRUS‐WENO) scheme is designed for solving the nonlinear degenerate parabolic equations on structured meshes. This new CRUS‐WENO scheme only uses the information defined on three unequal‐sized spatial stencils, obtains the optimal sixth‐order accuracy in smooth regions, and preserves the second‐order accuracy near strong discontinuities. This scheme can be applied to dispose the emergence of the negative linear weights and avoid the application of the mapped function. The corresponding linear weights can be artificially set to be any random positive numbers as well as their summation is one. The construction process of this scheme is very simple and can be easily extended to higher dimensions, since the new compact conservative formulation is applied to approximate the second‐order derivatives. The new CRUS‐WENO scheme uses narrower large stencil than that of the same order finite difference classical WENO schemes. Therefore, it has a better compactness and smaller truncation errors. Some benchmark numerical tests including the porous medium equation and the degenerate parabolic convection‐diffusion equation are performed to illustrate the advantages of this new CRUS‐WENO scheme.

An efficient fifth-order finite difference multi-resolution WENO scheme for inviscid and viscous flow problems

In this paper, a more efficient fifth-order finite difference multi-resolution WENO scheme is designed to solve the compressible inviscid and viscous flow problems based on the original multi-resolution WENO scheme presented in Zhu and Shu (2018). This new improved method inherits all the following features of the original multi-resolution WENO scheme (Zhu and Shu , 2018): (1) a series of unequal-sized central stencils are used to obtain the spatial reconstruction polynomials; (2) any positive numbers whose sum is one can be set as the linear weights; (3) smaller L1 and L∞ errors could be obtained for smooth problems; (4) it is easier to extend to the unstructured finite volume framework. However, different from the original multi-resolution WENO scheme, this improved method simplifies the reconstruction process, significantly improves the computational efficiency, and has greater engineering application potential. Numerical results show that the two types of fifth-order finite difference multi-resolution WENO schemes have similar results, but the CPU time of this new multi-resolution WENO scheme is about 0.7 times that of the original one. Moreover, the new fifth-order multi-resolution WENO scheme with a small increase in the computational cost shows less dissipation error than the classical WENO scheme (Jiang and Shu, 1996), and can capture more subtle flow structures for solving inviscid and viscous flow problems on the same mesh level. Several benchmark inviscid and viscous problems are illustrated to verify the above conclusions and the improved performance of this fifth-order multi-resolution WENO scheme.

Multi-resolution HWENO schemes for hyperbolic conservation laws

In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) [29]. The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) [19], in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes.

Novel functional weights for improving the third‐order WENO schemes

AbstractIn this article, we improved the nonlinear weight of the classical third‐order WENO scheme by using the limiter of the MUSCL scheme and got a new WENO scheme, the WENO‐F scheme, and developed three new nonlinear multiple‐parameter weight functions by improving the nonlinear weight of WENO‐JS, WENO‐Z, and WENO‐Z+ schemes. After selecting appropriate parameters of these new weight functions, three new high‐resolution schemes are comparing with three kinds of classical WENO schemes. Theoretical analysis and numerical results show that the new scheme with these multiparameter functional weights can achieve third‐order convergence in smooth regions. And the new scheme has lower dissipation and better resolution than the classical third‐order WENO schemes for smooth and discontinuous solutions.

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

In this paper we develop a family of very high-order central (up to 6th-order) non-oscillatory schemes for mixed-element unstructured meshes. The schemes are inherently compact in the sense that the central stencils employed are as compact as possible, and that the directional stencils are reduced in size therefore simplifying their implementation. Their key ingredient is the non-linear combination in a CWENO style similar to Dumbser et al [1] of a high-order polynomial arising from a central stencil with lower-order polynomials from directional stencils. Therefore, in smooth regions of the computational domain the optimum order of accuracy is recovered, while in regions of sharp-gradients the larger influence of the reconstructions from the directional stencils suppress the oscillations. It is the compactness of the directional stencils that increases the chances of at least one of them lying in a region with smooth data, that greatly enhances their robustness compared to classical WENO schemes. The two variants developed are CWENO and CWENOZ schemes, and it is the first time that such very-high-order schemes are designed for mixed-element unstructured meshes. We explore the influence of the linear weights in each of the schemes, and assess their performance in terms of accuracy, robustness and computational cost through a series of stringent 2D and 3D test problems. The results obtained demonstrate the improved robustness that the schemes offer, a parameter of paramount importance for and their potential use for industrial-scale engineering applications.

High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems

Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical L2 projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.

Arc Length-Based WENO Scheme for Hamilton–Jacobi Equations

In this article, novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton–Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.

Seventh and ninth orders characteristic-wise alternative WENO finite difference schemes for hyperbolic conservation laws

In this work, the characteristic-wise alternative formulation of the seventh and ninth orders conservative weighted essentially non-oscillatory (AWENO) finite difference schemes is derived. The polynomial reconstruction procedure is applied to the conservative variables rather than the flux function of the classical WENO scheme. The numerical flux contains a low order term and high order derivative terms. The low order term can use arbitrary monotone fluxes that can enhance the resolution and reduce numerical dissipation of the fine scale structures while capturing shocks essentially non-oscillatory. The high order derivative terms are approximated by the central finite difference schemes. The improved performance in terms of accuracy, essentially non-oscillatory shock capturing and resolution for the complex shocked flow with fine scale structures in the classical one- and two-dimensional problems is demonstrated. However, the inclusion of the high order derivative terms is prone to generate Gibbs oscillations around a strong discontinuity and might result in a negative density and/or pressure. Therefore, a positivity-preserving limiter [Hu et al. J. Comput. Phys. 242 (2013)] is adopted to ensure the positive density and pressure in the shocked flows with extreme conditions, such as Mach 2000 jet flow problem.

A new type of increasingly high-order multi-resolution trigonometric WENO schemes for hyperbolic conservation laws and highly oscillatory problems

In this paper, we investigate designing a new type of high-order finite difference multi-resolution trigonometric weighted essentially non-oscillatory (TWENO) schemes for solving hyperbolic conservation laws and some benchmark highly oscillatory problems. We only use the information defined on a hierarchy of nested central spatial stencils in a trigonometric polynomial reconstruction framework without introducing any equivalent multi-resolution representations. These new finite difference trigonometric WENO schemes use the same large stencils as the classical WENO schemes (Jiang and Wu, 1996; Shu, 2009), could obtain the optimal order of accuracy in smooth regions, and simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such multi-resolution trigonometric WENO schemes can be any positive numbers on condition that their summation is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference trigonometric WENO schemes. These new trigonometric WENO schemes are simple to construct and can be easily implemented to arbitrary high-order accuracy in multi-dimensions. Some benchmark examples including some highly oscillatory problems are given to demonstrate the robustness and good performance of these new trigonometric WENO schemes.

Efficient seventh order WENO schemes of adaptive order for hyperbolic conservation laws

In this article, new efficient seventh order adaptive WENO schemes are proposed for hyperbolic conservation laws. The accuracy of proposed schemes are comparable with existing versions of seventh order adaptive WENO scheme known as WENO-AO(7,3) and WENO-AO(7,5,3) [Balsara, Garain, and Shu, J. Comput. Phys., 326 (2016), pp. 780–804]. The accuracy of the new family of WENO schemes in resolving shocks or other discontinuities and high frequency waves are better than that of WENO-AO(7,3) and WENO-AO(7,5,3) schemes on a coarse mesh. In general, WENO schemes with adaptive order have the drawback that they involve computation of an extra smoothness indicator over the bigger stencil as compared to classical WENO schemes. In order to reduce the computational cost, a new simple smoothness indicator is proposed for the bigger stencil using the information available from smoothness indicators defined over smaller stencils and it is equal to original smoothness indicator upto O(Δx6) at truncation error level. The new WENO schemes of adaptive order developed using the new smoothness indicator gives comparable solution to other WENO schemes of adaptive order. We have proved that in each WENO reconstruction of adaptive order, nonlinear weights approach the linear weights and flux converges with order of seven, when the solution is smooth over the underlying stencil S07. Extensive numerical experiments are performed in one and two dimensions to validate the present algorithms.

Central WENO Schemes Through a Global Average Weight

A novel central weighted essentially non-oscillatory (central WENO; CWENO)-type scheme for the construction of high-resolution approximations to discontinuous solutions to hyperbolic systems of conservation laws is presented. This procedure is based on the construction of a global average weight using the whole set of Jiang–Shu smoothness indicators associated to every candidate stencil. By this device one does not to have to rely on ideal weights, which, under certain stencil arrangements and interpolating point locations, do not define a convex combination of the lower-degree interpolating polynomials of the corresponding sub-stencils. Moreover, this procedure also prevents some cases of accuracy loss near smooth extrema that are experienced by classical WENO and CWENO schemes. These properties result in a more flexible scheme that overcomes these issues, at the cost of only a few additional computations with respect to classical WENO schemes and with a smaller cost than classical CWENO schemes. Numerical examples illustrate that the proposed CWENO schemes outperform both the traditional WENO and the original CWENO schemes.

Reprint of: EBR-WENO scheme for solving gas dynamics problems with discontinuities on unstructured meshes

We develop the Edge-Based Reconstruction WENO scheme for solving Euler equations on unstructured meshes. It belongs to the class of edge-based schemes with quasi-1D reconstruction of variables. The scheme monotonization is provided by using a convex combination of three lower-order reconstructions of variables in a similar way as it is in the classical finite-difference WENO scheme. The new scheme damps oscillations near shocks on unstructured meshes and, due to its edge-based nature, requires rather low computational costs. The properties of the new scheme are demonstrated on several test problems.

A new class of high-order weighted essentially non-oscillatory schemes for hyperbolic conservation laws

In this paper, a class of even order WENO schemes, including fourth-order, sixth-order and eighth-order schemes, are presented in finite volume framework for hyperbolic conservation laws. Instead of two upwind stencils in the classical WENO reconstruction [10] for each cell interface, a common symmetrical stencil is used in the reconstruction of the variables at its both sides. For each cell interface, the 2rth order scheme shares the same number of cell averaged values with the (2r−1)th order classical WENO scheme. To suppress the spurious oscillation and improve the resolution in the region with discontinuities, a convex combination and a nonlinear weighting strategy are given with the unequal degree polynomials from the candidate sub-stencils. In smooth region, the current schemes achieve one order of improvement in accuracy compared with the classical WENO schemes. A variety of numerical tests are presented to validate the performance of the current schemes. Complex flow structures and discontinuities can be well resolved, and the robustness is as good as that of the classical WENO schemes.

EBR-WENO scheme for solving gas dynamics problems with discontinuities on unstructured meshes

We develop the Edge-Based Reconstruction WENO scheme for solving Euler equations on unstructured meshes. It belongs to the class of edge-based schemes with quasi-1D reconstruction of variables. The scheme monotonization is provided by using a convex combination of three lower-order reconstructions of variables in a similar way as it is in the classical finite-difference WENO scheme. The new scheme damps oscillations near shocks on unstructured meshes and, due to its edge-based nature, requires rather low computational costs. The properties of the new scheme are demonstrated on several test problems.

Central Weighted ENO Schemes for Hyperbolic Conservation Laws on Fixed and Moving Unstructured Meshes

We present a novel arbitrary high order accurate central WENO spatial reconstruction procedure (CWENO) for the solution of nonlinear systems of hyperbolic conservation laws on fixed and moving unstructured simplex meshes in two and three space dimensions. Starting from the given cell averages of a function on a triangular or tetrahedral control volume and its neighbors, the nonlinear CWENO reconstruction yields a high order accurate and essentially non-oscillatory polynomial that is defined everywhere in the cell. Compared to other WENO schemes on unstructured meshes, the total stencil size is the minimum possible one, as in classical point-wise WENO schemes of Jiang and Shu. However, the linear weights can be chosen arbitrarily, which makes the practical implementation on general unstructured meshes particularly simple. We make use of the piecewise polynomials generated by the CWENO reconstruction operator inside the framework of fully discrete and high order accurate one-step ADER finite volume schemes on fixed Eulerian grids as well as on moving Arbitrary-Lagrangian-Eulerian (ALE) meshes. The computational efficiency of the high order finite volume schemes based on the new CWENO reconstruction is tested on several two- and three-dimensional systems of hyperbolic conservation laws and is found to be more efficient in terms of memory consumption and computational efficiency with respect to classical WENO reconstruction schemes on unstructured meshes. We also provide evidence that the new algorithm is suitable for implementation on massively parallel distributed memory supercomputers, showing two numerical examples run with more than one billion degrees of freedom in space. To our knowledge, at present these are the largest simulations ever run with unstructured WENO finite volume schemes.

Схема EBR-WENO для решения задач газовой динамики с разрывами на неструктурированных сетках

Abstract We develop the Edge-Based Reconstruction WENO scheme for solving Euler equations on unstructured meshes. It belongs to the class of edge-based schemes with quasi-1D reconstruction of variables. The scheme monotonization is provided by using a convex combination of three lower-order reconstructions of variables in a similar way as it is in the classical finite-difference WENO scheme. The new scheme damps oscillations near shocks on unstructured meshes and, due to its edge-based nature, requires rather low computational costs. The properties of the new scheme are demonstrated on several test problems.